Quantum Algebras and Cyclic Quiver Varieties (1504.06525v2)

Abstract: The purpose of this thesis is to present certain viewpoints on the geometric representation theory of Nakajima cyclic quiver varieties, in relation to the Maulik-Okounkov stable basis. Our main technical tool is the shuffle algebra, which arises as the K-theoretic Hall algebra of the double cyclic quiver. We prove the isomorphism between the shuffle algebra and the quantum toroidal algebra U_{q,t}(sl_n^^), and identify the quotients of Verma modules for the shuffle algebra with the K-theory groups of Nakajima cyclic quiver varieties, which were studied by Nakajima and Varagnolo-Vasserot. The shuffle algebra viewpoint allows us to construct the universal R-matrix of the quantum toroidal algebra U_{q,t}(sl_n^^), and to factor it in terms of pieces that arise from subalgebras isomorphic to quantum affine groups U_{q}(gl_m^), for various m. This factorization generalizes constructions of Khoroshkin-Tolstoy to the toroidal case, and matches the factorization that Maulik-Okounkov produce via the stable basis in the K-theory of Nakajima quiver varieties. We connect the two pictures by computing formulas for the root generators of U_{q,t}(sl_n^^) acting on the stable basis, which provide a wide extension of Murnaghan-Nakayama and Pieri type rules from combinatorics.